CDIP v3.1 Geometry-to-Occupancy Bridge Theorem I A Local Combinatorial Transport Bound Under Negative Ollivier–Ricci Curvature

This paper studies the missing interface between discrete geometric diagnostics and stochastic shutdown laws in the CDIP v3.1 theorem line. The target bridge is a lower bound on risk occupancy: if a Markov kernel evolves on a graph region exhibiting sufficiently negative Ollivier--Ricci curvature, can one conclude that a designated risk-edge set is visited with uniformly positive probability? In full generality, the answer is negative: Wasserstein expansion alone does not control the mass of an arbitrary bad set. The paper therefore develops a local combinatorial theorem under a controlled graph-kernel regime.

The main result shows that, on a locally supported Markov kernel over a simple unweighted graph, sufficiently negative Ollivier--Ricci curvature along an edge forces the existence of at least one outward branch atom carrying a strictly positive amount of transition mass, provided a minimal atom-mass lower bound is available. If, in addition, the designated risk-edge set is aligned with outward branches, then the risk-edge occupancy of the local transition kernel admits a uniform positive lower bound. This yields exactly the conditional quantity needed by the TEANT average-occupancy shutdown theorem.

The paper also clarifies the sharp boundary of the method. A general bridge theorem from Wasserstein expansion to arbitrary risk-set mass is not proved and should not be claimed. The present result instead isolates a precise, publishable local regime in which geometry-to-occupancy lower bounds are mathematically valid. As throughout the CDIP v3.1 theorem line, the interpretation remains NI-2: alerts-only, no threshold changes, no gate changes, no verdict lifting, and no write-back semantics.